# Rodion Kuzmin

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Template:Infobox scientist Rodion Osievich Kuzmin (Russian: Родион Осиевич Кузьмин{{#invoke:Category handler|main}}, Nov. 9, 1891, Riabye village in the Haradok district – March 23, 1949, Leningrad) was a Russian mathematician, known for his works in number theory and analysis.[1] His name is sometimes transliterated as Kusmin.

## Selected results

${\displaystyle x={\frac {1}{k_{1}+{\frac {1}{k_{2}+\cdots }}}}}$
is its continued fraction expansion, find a bound for
${\displaystyle \Delta _{n}(s)=\mathbb {P} \left\{x_{n}\leq s\right\}-\log _{2}(1+s),}$
where
${\displaystyle x_{n}={\frac {1}{k_{n+1}+{\frac {1}{k_{n+2}+\cdots }}}}.}$
Gauss showed that Δn tends to zero as n goes to infinity, however, he was unable to give an explicit bound. Kuzmin showed that
${\displaystyle |\Delta _{n}(s)|\leq Ce^{-\alpha {\sqrt {n}}}~,}$
where C,α > 0 are numerical constants. In 1929, the bound was improved to C 0.7n by Paul Lévy.
${\displaystyle 2^{\sqrt {2}}=2.6651441426902251886502972498731\ldots }$
is transcendental. See Gelfond–Schneider theorem for later developments.
${\displaystyle \sum _{n\in I}e^{2\pi if(n)}\ll \lambda ^{-1}.}$

## Notes

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